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Mirrors > Home > MPE Home > Th. List > ovresd | Structured version Visualization version GIF version |
Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ovresd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
ovresd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑋) |
Ref | Expression |
---|---|
ovresd | ⊢ (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovresd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
2 | ovresd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑋) | |
3 | ovres 7598 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 × cxp 5686 ↾ cres 5690 (class class class)co 7430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-xp 5694 df-res 5700 df-iota 6515 df-fv 6570 df-ov 7433 |
This theorem is referenced by: sscres 17870 fullsubc 17900 fullresc 17901 funcres2c 17954 rngchom 20639 ringchom 20668 rhmsubclem4 20704 irinitoringc 21507 psmetres2 24339 xmetres2 24386 prdsdsf 24392 xpsdsval 24406 xmssym 24490 xmstri2 24491 mstri2 24492 xmstri 24493 mstri 24494 xmstri3 24495 mstri3 24496 msrtri 24497 tmsxpsval 24566 ngptgp 24664 nlmvscn 24723 nrginvrcn 24728 nghmcn 24781 cnmpt1ds 24877 cnmpt2ds 24878 ipcn 25293 caussi 25344 causs 25345 minveclem2 25473 minveclem3b 25475 minveclem3 25476 minveclem4 25479 minveclem6 25481 ftc1lem6 26096 ulmdvlem1 26457 abelth 26499 cxpcn3 26805 rlimcnp 27022 hhssnv 31292 madjusmdetlem3 33789 qqhcn 33953 qqhucn 33954 ftc1cnnc 37678 ismtyres 37794 isdrngo2 37944 naddcnffo 43353 |
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